"He forgot there was an important meeting today."
This sentence seems like it is acceptable in terms of phrasing. However, it is close to "Today, he forgot there was an important meeting". What you are doing with 今天 is that the time of the "meeting" is not indicated, so that the meeting can take place in any of the past time, say yesterday or last week. Instead, you are making "he" does the job "today", forgetting.
Do be careful of how you combine characters together since in practice, Chinese natives are often prosy and descriptive with sentences.
I shall bear that in mind, thanks. But I would have thought that putting the date earlier merely confirms both the forgetting and the meeting were both today. One lives & learns. One thinks one has learnt something of the grammar, then along comes something to trip one up.